I'm not sure about books but there is Castle, Father Brown and Dr Blake currently on TV. In Australia there is also the Miss Fisher Mysteries that are rather fab and I have to buy the DVD of.
So no it's not dead. Most are set in the past but Castle is modern. Father Brown, a priest, and Dr Blake, a police surgeon/doctor, in a sleepy backwater could easily be updated.
How 'feasible' is a story in this style, especially in the modern era?
It's all in how you execute the story. If you write it well with engaging characters, an intriguing plot, and you keep it true then it's very feasible indeed.
In general, though, Detective stories are a hard sell these days. That could be because most of the ones being written today are are tired retreads of better books that came before.
Father Brown and Phryne Fisher both made their debut in book series. Father Brown -- G K Chesterton; Phryne Fisher -- Kerry Greenwood.
There is no such thing as a dead genre, or a dead sub-genre. All such genres are only sleeping, waiting for a really good writer to wake them up.